Dynamic alpha-invariants of del Pezzo surfaces

TL;DR

This paper computes the alpha-invariant for del Pezzo surfaces with edge singularities and establishes conditions for the existence of Kähler--Einstein metrics with cone angles along curves.

Contribution

It provides explicit calculations of Tian's alpha-invariant for del Pezzo surfaces and proves existence results for Kähler--Einstein metrics with edge singularities.

Findings

Computed alpha-invariants for all smooth del Pezzo surfaces.

Established existence of Kähler--Einstein metrics with cone singularities for certain angles.

Provided lower bounds for the supremum of cone angles allowing such metrics.

Abstract

For every smooth del Pezzo surface $S$, smooth curve $C\in|-K_{S}|$ and $\beta\in(0,1]$, we compute the $\alpha$-invariant of Tian $\alpha(S,(1-\beta)C)$ and prove the existence of K\"ahler--Einstein metrics on $S$ with edge singularities along $C$ of angle $2\pi\beta$ for $\beta$ in certain interval. In particular we give lower bounds for the invariant $R(S,C)$, introduced by Donaldson as the supremum of all $\beta\in(0,1]$ for which such a metric exists.